In this paper, we study a compensation scheme for linear sequencing situations proposed by Curiel et al. (1989). Instead of focusing on the allocation of the total cost savings among players, we concentrate on the actual monetary transfers arising from each neighboring switch. First, inspired by the split core introduced by Hamers et al. (1996), we introduce a compensation scheme that accounts for the losses incurred by players who are moved to later positions in the queue. Second, we propose two properties to characterize the compensation scheme, namely the Compensation Balance property and the Proportional Balanced Net Payoff property. The former implies that for any neighboring switch of two players, the compensation loss of the forward-moving player is exactly balanced by the compensation gain of the backward-moving player. The latter implies that for any two inverse players, their net payoffs are proportional to their bargaining abilities. Moreover, we construct a cooperative compensation game and demonstrate that the Shapley value of this game coincides with the proposed compensation scheme. Finally, we extend the compensation scheme from linear sequencing situations to a broader class of sequencing situations with general cost structures by proposing a path-based compensation scheme.
This paper proposed a modified version of whale optimization algorithm (MVWOA), which incorporates four critical components. Firstly, a dynamic reverse learning strategy is designed to improve the uniformity of the initial population with better location in the optimization process. Secondly, to mitigate premature convergence, a Lévy flight mechanism with an optimized pa factor is proposed to increase population diversity over the course of iterations. Thirdly, an information acquisition and sharing strategy facilitates inter-agent communication within the population for accelerating the convergence rate. Finally, a differential evolution strategy is integrated to improve the efficiency of approaching the obtained solutions. Besides that, unmanned aerial vehicle (UAV) path planning models in two-dimensional and three-dimensional are also bulit to measure the proposed algorithm in practical. The performance of MVWOA is evaluated on the CEC2022 benchmark set as well as UAV path planning problems. Simulation results indicate that the proposed algorithm outperforms some state-of-the-art algorithms considering on accuracy and convergence, demonstrating its potential for solving applications.
In today’s economy, pricing strategies and environmentally friendly practices become progressively interconnected and essential for business success. This research aims to explore how different strategies influence optimal decisions in a two-period closed-loop supply chain (CLSC) where demand depends on both the product’s price and its green level. The proposed CLSC consists of one manufacturer and one retailer. In the forward channel, the manufacturer sells products to consumers through a conventional retail channel, while in the reverse channel, used goods are collected and returned to the manufacturer for remanufacturing. To maximize profits, both parties employ two decision-making strategies: pre-announcing and dynamic (responsive) approaches. Additionally, two specific scenarios are analyzed where the manufacturer sells exclusively non-green products or only green products during the second period. The analysis reveals that the pre-announcing strategy yields higher profit with lower selling price, while the dynamic strategy results in a higher green level for the product. Furthermore, while the retailer’s profit, the overall supply chain profit, and the product return rate all increase when the supply chain shifts from green to non-green products, the manufacturer favors focusing on green products in order to maximize his own profit.
With the rapid development of the Internet and third-party logistics, many consumers are increasingly relying on online channel to buy their products as opposed to traditional retail channels. In recent years, the dual-channel fresh produce supply chain (FPSC) dominated by the supplier with an online sale channel and an offline sale channel is an important sale mode. For the FPSC, achieving channel coordination and resolving the channel conflict between online channel and offline channel is crucial for its implementation. However, the coordination problem of the FPSC composed of one supplier and one retailer, in which the supplier undertakes corporation social responsibility (CSR) and the retailer performs freshness-keeping effort (FKE), has been seldom studied. The channel coordination of such a dual-channel FPSC is investigated in this study. For the centralized problem, firstly the profit of the FPSC is proved to be concave with respect to the CSR investment, the FKE and the online and offline sale prices under certain conditions. We show that the optimal sale prices and the profit of the FPSC are higher than those of the FPSC without CSR and FKE. Subsequently, the equilibrium solution of FPSC under the wholesale price contract is solved to get baseline profits. Then, a mixed contract combining two-tariff contract and channel profit-sharing contract is proposed to coordinate the FPSC and achieve Pareto improvement of the profit and the welfare of consumer. Lastly, two numerical examples are used to illustrate the correctness of theoretical analysis, and are employed to investigate the impact of CSR effect factor and freshness sensitivity on the profit, the CSR investment and FKE. The managerial insights for improving the FPSC are also presented.
Traditional data envelopment analysis (DEA) models are limited by their inherent assumption of convexity, which hampers their ability to effectively approximate non-convex production possibility sets (PPS). While artificial intelligence (AI) methods offer greater flexibility by overcoming convexity constraints, they can be adversely influenced by decision-making units that fail to maintain monotonicity. To address this challenge, we propose a novel hybrid approach that integrates the Free Disposal Hull (FDH) method for preprocessing data with the Radial Basis Function (RBF) network, an AI algorithm, to estimate production frontiers and efficiency scores. By combining the nonparametric capabilities of FDH with the smooth surface approximation of RBF network, this method harnesses the strengths of both AI and optimization. A simulation based on the Cobb–Douglas production function demonstrates the superiority of the FDH-RBF approach, particularly in scenarios exhibiting increasing marginal returns, where it outperforms traditional methods in terms of accuracy. Applying this method to estimate the production frontiers of Chinese cities reveals a potential ‘medium-sized efficiency trap,’ where medium-sized cities consistently underperform. These findings illustrate the value of integrating AI and optimization models for complex production scenarios, offering more accurate and adaptable solutions.
The complex fuzzy set and (r, s, t)-Spherical fuzzy set serve as valuable extensions of various fuzzy structures, including fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, q-rung orthopair fuzzy sets, spherical fuzzy sets, and T-spherical fuzzy sets. In modelling some real-world problems, using complex values, which play a significant role in representing multidimensional data, is often more advantageous than using crisp values. In this study, set and arithmetic operations are redefined among two Crst-SFSs, and some properties regarding these operations are discussed. In addition, a Crst-SFN’s accuracy and score functions are redefined. Additionally, distance measure (DM)s using Hausdorff, Euclidean, and Hamming measures between two Crst-SFSs are put forward. Furthermore, a multi-criteria group decision-making (MCGDM) technique based on TOPSIS method using proposed score function and DMs is developed. A case study is presented to show how the suggested MCGDM approach can be used. Finally, a comparison is made among the current and novel methods.
Pickup and delivery operations with cross-docking play a critical role in urban freight logistics by enabling rapid consolidation and synchronization of goods flows. The increasing adoption of electric vehicles in such systems is motivated by environmental targets but introduces operational challenges related to limited driving range, battery charge feasibility, and charging constraints under strict time-window requirements. This study investigates whether crossdock-based pickup and delivery systems can be operated feasibly and efficiently when vehicle routing decisions must also satisfy battery-related constraints. To address this question, a mixed-integer linear programming model is developed that integrates battery charge dynamics and opportunistic recharging directly into the crossdock handling process, ensuring energy feasibility across multiple routing phases. Numerical experiments show that, while the use of electric vehicles leads to moderately higher operating costs compared to conventional fleets, feasible and operationally consistent solutions can be achieved without additional routing detours. Overall, the proposed formulation provides an optimization framework for assessing the trade-offs between energy constraints, synchronization requirements, and routing efficiency in sustainable urban logistics systems.
In today’s market, shelf space and expiration dates significantly impact an item's demand. This study develops an inventory model that incorporates shelf space and expiration rates, accounting for varying lead time scenarios. Consumer preference for freshness is increasing under the current EOQ model, which drives investments in preservation technology to enhance the freshness of perishable goods. However, this often leads to increased carbon emissions. Companies can mitigate the risk of expiring products by running targeted advertising, offering discounts, and adjusting inventory levels based on demand forecasts and expiration dates. Thus, this study creates a model that addresses demand influenced by advertisement, stock levels, and expiration rates, considering three lead time scenarios: (i) items arriving before stock depletion, avoiding shortages; (ii) items arriving just in time, maintaining inventory levels; and (iii) items arriving after stock depletion, leading to backlogs. To account for uncertainty in parameters, the study employs cylindrical and triangular neutrosophic numbers. Numerical examples and sensitivity analyses are conducted to evaluate the effects of parameter variations on the model’s performance. The overall expenses decrease by 20
Although data envelopment analysis (DEA) has been widely used to address fixed cost allocation problems from an efficiency perspective, such allocations may lead to imbalanced cost burdens and unequal payoffs among DMUs, thereby raising fairness concerns. This paper proposes an approach to determine fixed cost and common revenue allocations that jointly maximize fairness-based payoffs and efficiencies for all decision-making units (DMUs) under dual fairness concerns. The dual concerns involve fairness-based payoffs capturing DMUs’ perceived fairness and the fairness of fixed cost and common revenue allocation. The former gives rise to a possible set of fairness-based Nash equilibrium fixed cost and common revenue allocations maximize all DMUs’ fairness-based payoffs. To ensure that all DMUs achieve maximum efficiency after allocation, a common-weights DEA model is integrated with the equilibrium possible set, resulting in a non-empty possible set of allocations. Furthermore, three fairness principles of allocations are proposed: one emphasizing fairness in fixed cost allocation, one focusing on fairness in common revenue allocation, and one balancing the trade-off between the two. Based on these principles, three fairness-oriented fixed cost and common revenue allocations are derived from the non-empty possible set. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed methodology. The results show that the proposed approach not only incorporates fairness considerations but also ensures that both the efficiencies and fairness-based payoffs of all DMUs are maximized.
Despite artificial intelligence (AI) algorithms often outperforming human predictions, decision-makers frequently discount or override forecasts generated by AI systems. This behavioral tendency, often referred to as AI aversion, has important implications for wholesale price setting and contract negotiations between a retailer and a manufacturer. In this study, we examine two prevalent contract forms—a wholesale price contract and a two-part tariff contract—and use the Nash bargaining solution to investigate how a retailer’s aversion to algorithm-generated forecasts affects wholesale prices and expected profits. Our analysis shows that the retailer prefers the wholesale price contract, whereas the manufacturer prefers the two-part tariff contract. Moreover, the effect of AI aversion depends on the discrepancy between the retailer’s own demand estimate and the AI forecast. When the retailer’s own estimate exceeds the AI forecast, stronger AI aversion raises the bargained wholesale price and can increase both parties’ expected profits. When the retailer’s own estimate is lower than the AI forecast, stronger AI aversion reduces expected profits for both parties. Numerical experiments further illustrate these results. By explicitly modeling AI aversion as the retailer’s behavioral discounting of an algorithm-generated forecast, this study provides new insight into trust, bargaining, and contract design in AI-assisted supply chains.
The growing public environmental awareness and green demands are prompting the government and manufacturers to emphasize green manufacturing and operations. Due to the growth of e-commerce, manufacturers are increasingly entering the retail market to sell products directly to consumers. The green supply chain in this paper is integrated with manufacturer encroachment strategies and government subsidy policies. We consider two manufacturer encroachment strategies (i.e., centralized and decentralized encroachments) and two government subsidy policies (i.e., green degree subsidy and R D cost subsidy). In these settings, we construct four Stackelberg game models and derive the optimal solutions. First, the positive effects of government subsidies and consumer green awareness on green production are demonstrated. Notably, increasing product substitutability harms the manufacturer but favors product greenness and the retailer’s profitability. Second, under each government subsidy policy, decentralized encroachment outperforms centralized encroachment in improving product greenness and each party’s profits. Third, the ratio of environmental performance and purchasing price is examined. We conclude that consumers stand to gain from the enhanced subsidy coefficient, and that heightened consumer green awareness is positively correlated with this benefit. We also find that, under a fixed government subsidy on expenditure, when consumers have sufficiently high green awareness, the R D cost subsidy policy makes products greener. Nevertheless, the green degree subsidy policy benefits each member of the supply chain. Finally, the robustness of our qualitative findings is verified by numerical modeling and analysis, and some essential managerial countermeasures are also proposed.
Live-streaming e-commerce has gained immense popularity, and choosing an effective live-streaming showcasing strategy is crucial. This paper constructs a dual-channel retailing system under three live-streaming showcasing strategies: influencer live-streaming, merchant live-streaming, and virtual character live-streaming. We employ the Stackelberg game and derivative optimization to investigate the optimal live-streaming showcasing strategy and analyze the impact of consumer returns. The main conclusions are as follows: (i) There is no universally best or worst live-streaming showcasing strategy, and the strategy selection is related to the fixed payment from the online retailer to the influencer-streamer and the R D cost coefficient for the virtual-streamer. (ii) The optimal refund policy is highly dependent on the return cost. A partial refund policy is preferable under the influencer live-streaming strategy when the return cost is high, whereas a full refund policy becomes more advantageous for both merchant live-streaming and virtual character live-streaming strategies. (iii) The return cost has a differential impact on optimal decisions under different refund policies. Under the merchant live-streaming strategy, adopting a full refund policy leads to both the price and promotional level declining as the return cost increases. Conversely, adopting a partial refund policy results in both the price and promotional level rising as the return cost increases.
The emergence of Buy-Now-Pay-Later (BNPL) services has significantly boosted consumers’ willingness to make purchases in recent years. This paper explores the impact of BNPL installment payments on retailer replenishment decisions in a multi-retailer inventory replenishment model, and studies the problem of cost allocation among retailers from a cooperative game perspective. We first explore the optimal ordering policy to minimize the total cost in both independent and joint replenishment scenarios. We find that joint replenishment can reduce the total cost for retailers and shorten their optimal ordering cycle compared to individual ordering. Then, to allocate the cooperative costs among retailers participating in the joint replenishment, we introduce a joint replenishment game with BNPL installment payments and show that this game is concave. Based on this, we propose an easy-to-compute proportional allocation rule which is stable (in the sense of core) and consistent (can be reached through population monotonic allocation scheme). Finally, we use a numerical example to compare our proposed allocation rule with the classical Shapley value. The results indicate that the proposed rule is more favorable to retailers with smaller annual demands for the product, whereas the Shapley value favors retailers with larger demands.
Efficient terminal management relies heavily on the synchronized scheduling of berth space and the assignment of operational equipment. This study addresses the integrated Berth Allocation Problem (BAP) and machine assignment problem through the introduction of a novel machine-pattern modeling approach. In contrast to traditional formulations that rely on individual machine-allocation variables, the proposed framework utilizes machine patterns to represent resource availability. This approach significantly reduces model complexity and enables the simultaneous allocation of multiple machine types within a continuous-time framework. Two Mixed-Integer Linear Programming (MILP) formulations are presented, one at the quay level and another at the berth level, with the primary objective of minimizing the total weighted sum of vessel waiting and handling times. To address the computational challenges of real-world port operations, a dedicated heuristic algorithm is developed for large-scale instances. Computational experiments demonstrate the effectiveness of these methods; the heuristic is capable of solving instances involving up to 100 berths and 500 vessels in under three seconds. Furthermore, the heuristic solutions maintain high quality, averaging within 9
The scheduling of flow shops with multiple dedicated machines, a problem referred to in the literature under various names such as Hybrid Flow Shop with Dedicated Machines (HFSDM), Flow Shop with Parallel Dedicated Machines, Differentiation Flow Shop, or Flow Shop with a Critical Machine, presents a complex optimization challenge encountered in many real-life applications. Given its strong relevance in both theory and practice, this problem has attracted considerable attention in the literature. This growing interest underlines the need for a more detailed and structured analysis, which this paper aims to provide. We begin by examining the main theoretical aspects of the HFSDM problem, with a particular focus on its structural characteristics and computational complexity, in order to build a clear foundation for the study. We then propose a classification of existing works according to the solution approaches adopted, allowing for a more organized and readable overview of the field. It succinctly explores and examines several configurations of the HFSDM scheduling problem, each addressing distinct constraints, assumptions, and optimality criteria. The paper concludes by identifying emerging research opportunities and suggesting promising directions for future work in the field of Hybrid Flow Shop with Dedicated Machines.
In this paper, optimal decisions are searched for both the wholesaler and the retailer in a three-layer business chain of coastal biomass. The biomass deteriorates over time, and hence, its freshness decreases dynamically. The expiration period of the biomass is a fuzzy parameter, and the freshness of units is expiration-period dependent, so it is also fuzzy. The rate of selling of the biomass is influenced by the freshness of units and unit selling price, so it is also fuzzy. Depending upon the unit purchase cost, the retailer optimizes the retail price function and order volume, and knowing this order volume, the wholesaler fixes his/her order volume. The models are formulated as leader-follower Stackelberg game (SG) problems and coordinated decision-making for the decision-makers, i.e., the model is studied under three distinct scenarios: the wholesaler-retailer Stackelberg game (WRSG), the retailer-wholesaler Stackelberg game (RWSG), and the coordination scenario (CS). While the wholesaler determines the wholesale price and order volume, the retailer fixes the order quantity, ordering time, and retail price to optimize the respective profits. The model is mathematically represented with the help of fuzzy differential equations and fuzzy Riemann integration. At first, a fuzzy goal of the fuzzy average profit of the retailer is determined. Then the credibility measure of the retailer’s average profit concerning this goal is optimized for the marketing decisions. A simulation process is followed to determine this fuzzy measure. The efficiency of the Artificial Bee Colony (ABC) algorithm is enhanced by incorporating its search in multiple dimensions and is named MDABC. It is tested and trusted using CEC2017 and CEC2022 datasets and is applied in a nested way to determine the decisions for the decision-makers. Numerical illustrations and parametric studies are presented to analyse the models, and it is established that the coordinated decision is better compared to both Stackelberg game decisions. However, the WRSG is more realistic compared to RWSG.
This comprehensive review aims to address a complex part of operations research, the nurse rostering problem. It is a critical piece of healthcare management, which affects the operational efficiency of hospitals and, at the same time, the quality of patient care. Given the dynamics of the diversity in the healthcare settings and needs, optimising the nurse staff schedule not only improves the care delivery, but also affects staff satisfaction. This review covers a wide range of techniques, including heuristics, meta-heuristics, such as simulated annealing, variable neighborhood search, genetic algorithms, population-based meta-heuristics, such as plant propagation algorithm, bee colony optimisation, and differential evolution, and advanced approaches, like hyper-heuristics, stochastic programming, and hybrid approaches. Additionally, the study explores mathematical optimisation techniques, including integer programming, mixed integer programming, and branch-and-price algorithms, with a special focus on hybrid methods, which combine meta-heuristics with exact optimisation approaches. By analysing key and recent studies, the review shows that hybrid approaches, particularly those integrating meta-heuristics with mathematical models, appear to be the most effective in addressing the complexities of nurse rostering (NR). Future directions, on the other hand, indicate that the growing integration of machine learning and real-time scheduling systems can provide further adaptability and scalability for NR solutions.
On-time delivery is becoming increasingly crucial for industrial companies, due to their customers’ need for deliveries by specific dates. This paper investigates parallel machine scheduling problems with sequence-dependent setup times, a combinatorial challenge that has gained significant attention due to its practicality and relevance in real-world applications. Aiming to reduce total weighted tardiness, we introduce a mixed integer linear programming model and an effective iterated greedy approach with a strategic reconstruction operator. We evaluate the performance of our methods by comparing it with six well-established and related approaches. Our experiments, using a benchmark set of 900 instances, validate that the introduced iterated greedy algorithm consistently produces high-quality solutions.
This article deals with an economic order quantity (EOQ) inventory model with varying numbers of customers and real-time dependent demand under a neutrosophic environment. First of all, we consider a cost minimization classical EOQ model and then solve it by using the calculus approach. But in practice, there exists a competitive atmosphere where some parameters of the model are assumed to be flexible in nature, and they follow the three-valued logic like a neutrosophic set. Thus, considering neutrosophic model, we extend this problem into a matrix game problem. To solve the model, we utilize a new de-neutrosophication method via the max–min approach of the matrix game, followed by a net aggregated score of neutrosophic elements alone. A new solution algorithm has also been developed for numerical computation over a case study dataset. A comparative numerical analysis has been done to show the novelty of the proposed approach under the recent five existing methods on neutrosophic decision-making models. Our findings reveal that the inventory system cost differs significantly with respect to the existing state-of-arts. However, the standard score (z-score) statistical interpretations show, for the cases of the max–min matrix game with aggregation operator, the mean and standard deviation of inventory system cost are assumed to be 1240.64 and81.78, respectively. But it becomes 1327.51 and59.97 for the cases of other methods. The corresponding z-scores are − 1.975 and − 1.245 about the primal (initial) solution. The z-score of the numerical outputs obtained by the new method shows the novelty and hence validates the proposed approach. Finally, the managerial insights, advantages, limitations, and a conclusion have been incorporated, followed by a scope of future work.
Existing studies on coal-power supply chain coordination under cap-and-trade mainly assume deterministic carbon prices and relatively simple contract structures, leaving the joint design of operational coordination and carbon-price risk allocation under stochastic permit prices largely unexplored. This paper studies a two-echelon coal-power supply chain regulated by cap-and-trade when the carbon price is stochastic. Demand is linear and each tier can reduce its emission factor through costly abatement. We first derive closed-form Stackelberg equilibria under decentralised decision-making and the closed-form centralised system optimum as a benchmark. Under risk neutrality, carbon-price volatility does not change the optimal quantity or abatement decisions, but it magnifies profit risk through the chain’s net carbon position. We then propose a hybrid contract that combines net carbon position sharing with a two-part tariff. The contract coordinates the decentralised chain to the centralised benchmark and enables a full range of Pareto-improving profit allocations. To capture risk aversion, we further develop a distributionally robust mean-variance framework with a bounded carbon-price variance. The centralised policy trades off expected profit against exposure to carbon-price uncertainty, and the hybrid contract can implement this policy while sharing risk between members. Numerical experiments illustrate how the mean and volatility of carbon prices, abatement efficiencies, and allowance endowments shape decisions, risk sharing, and profits, and we discuss managerial implications for supply chains operating under carbon markets.
